Ramsey数和临界Ramsey数问题的研究

Ramsey理论是极值图论的重要分支，其研究意义大，涉及面广，难度高。Ramsey理论揭示了图的复杂性，改变了人们对有限离散结构复杂性的认识，其中基于Ramsey数问题建立的随机图为实际网络提供了有效的数学模型。本项目将利用组合结构、代数构造和概率方法研究Ramsey数和临界Ramsey数问题，主要研究内容包括：（1）研究Wenger图、Paley图和其他拟随机图的结构和性质，估计涉及偶圈和稠密图的Ramsey数；（2）研究Ramsey图中的剩余子图结构，估计一些常见图（如圈、路径等）的临界Ramsey数和边Ramsey数。.本项目提出了关于临界Ramsey数的新问题，给出利用临界Ramsey数估计边Ramsey数的新方法。本项目的目标是刻画临界Ramsey数所需极图的结构，估计图的Ramsey数和临界Ramsey数，丰富Ramsey理论问题的研究。

Ramsey theory is an important area in extremal graph theory that contains many hard problems, which are related to other areas tightly. Ramsey theory reveals the complexity of graphs, especially the complexity of finite and discrete structures. To estimate the Ramsey functions, random graphs have been established and they provide a powerful mathematical model for real networks. Our project will use combinatorial structure, algebraic construction and probabilistic method to study Ramsey number and critical Ramsey number problems, including: (1) Using Wenger graphs, Paley graphs and other quasi-random graphs to estimate the Ramsey numbers involving even cycles and dense graphs; (2) Using the redundant subgraphs in Ramsey graphs to bound the critical Ramsey numbers and the size Ramsey numbers of graphs, such as cycles and paths..Our project will propose some problems on the critical Ramsey number, implying a new way to bound the corresponding size Ramsey number. Our aim is to consider the Ramsey graphs and redundant subgraphs, determine or bound some Ramsey functions and some critical Ramsey functions, find the extremal graph structures for such functions, and enrich the research on Ramsey theory problems.